Honestly, more research is needed. Some research has already been done in the past 10 years, but not enough to really make solid conclusions. For example, there's no research to support the "10% of body weight + 1 pound" rule that you'll find all over the internet (example).
With kids, there's also some concern about the safety of weighted devices. The Live Science article Weighted Blankets: Harmless for Adults, Potentially Dangerous for Kids mentions two deaths that were linked to weighted blankets. (Oxford Health NHS has some guidance for use here.)
The Live Science article does a good job at citing (and linking to) studies that have been done. The first one "showed only that weighted blankets were safe for people to use, not whether they are effective in improving people's health":
In the earliest of those studies, published in 2008 in the journal Occupational Therapy in Mental Health, researchers asked 33 adults to rest under 30-lb. (13.6 kg) blankets for 5 minutes. They found that 33 percent showed a greater drop in skin conductance — a measure of arousal that is based on miniscule differences in the amount of sweat on the skin — with a weighted blanket than without. Nineteen participants said they felt more relaxed with the blanket, while eight said they felt equally relaxed either way, and three said they felt more anxious under the blanket.
The researchers noted that the results suggested that just laying down was enough to physiologically relax most participants enough that any additional benefit of the weighted blanket disappeared.
The second study was done with mental patients:
In another study, published in 2012 in the journal Australasian Psychiatry, researchers found that the use of weighted blankets in an inpatient mental psychiatric unit decreased the amount of distress the patients reported feeling, and how anxious they felt toward their health care providers. But the blankets didn't seem to improve objective measurements like aggression or the hospital's need to seclude the patients away from others, the study said.
Only one of the studies dealt with sleep:
In a third study, published in 2014 in the journal Pediatrics, researchers tested using weighted blankets to help kids with autism who had problems sleeping. The study included a "placebo" blanket, containing light plastic beads to mimic the texture of the weighted blankets, which were filled with steel beads. The researchers measured how long it took the kids in the study to fall asleep, how often they woke during the night and how long they slept in total, using both reports by parents and activity monitors that the children wore.
"We found nothing," said Dr. Paul Gringras, the head of the Children's Sleep Medicine Unit at the Evelina London Children's Hospital, who led the study. "We found no difference. It was kind of disappointing."
The last study is the most promising in terms of support for weighted blankets:
In the fourth study, published in 2016 in the Journal of the Formosan Medical Association, researchers suggested that the blankets may help people in stressful circumstances. The researchers found that the study participants who wore a weighted blanket during wisdom tooth extraction showed enhanced activity in the branch of the nervous system that takes over in times of low stress.
The results suggest that "deep-pressure input may be an appropriate therapeutic modality" for people in stressful conditions, the researchers concluded.